So we've discovered that \(S_2(\Gamma _0(N), \mathbb {F}_2)^{\operatorname{Katz}}[\mathfrak {m}]\)  is at most 2 dimensional, spanned by \(Vf\)  and \(Af\) . Here, \(V\)  acts as \(V\left(\sum _{n=1}^{\infty }a_nq^n\right)=\sum _{n=1}^{\infty }a_nq^{2n}\)  on power series, so that \(Vf=g\) , and can either be a weight-doubling operator, as used in [1]}, or a level-doubling operator. Additionally, \(Af\)  is the multiplication of \(f\)  with the Hasse invariant \(A\) , which preserves \(q\) -expansions. We can hence calculate the action of \(T_2\)  on this space: we know that \(T_2\)  acts in weight 2 via \(U+2V\) , where \(U\left(\sum _{n=1}^{\infty }a_nq^n\right)=\sum _{n=1}^{\infty }a_{2n}q^n\) , and in weight 1 as \(U+\langle 2\rangle V\)  with \(\langle 2\rangle \)  the diamond operator, which is identically 1 on mod 2 forms. Further, we can compute that \(UVf=Af\) , as \(V\)  doubles each exponent and \(U\)  halves it.
